After computing max flow, the residual graph has flow on some edges. For each unit of flow, trace a path from to using edges with positive flow. You are reconstructing the paths. Start at , follow any edge with flow to the next vertex, and repeat until you reach .
Subtract from the flow on each edge you use. Repeat this process until no more flow remains. Each trace gives you one edge-disjoint path. The subtraction ensures paths do not overlap on edges.